Geometry Indiana Academic Standards Crosswalk 2014 2015 The Process Standards demonstrate the ways in which students should develop conceptual understanding of mathematical content and the ways in which students should synthesize and apply mathematical skills. These process standards should be embedded in and taught with all content standards. See pages 10 and 11 for more information. Indiana College and Career Ready Standards 2013-2014 Academic Standards Comments Content Standard Standard/Indicator G.LP.1: Understand and describe the structure of and IAS G.8.6: Identify and give examples of undefined relationships within an axiomatic system (undefined terms, axioms, and theorems, and inductive and terms, definitions, axioms and postulates, methods of deductive proofs. reasoning, and theorems). Understand the differences among supporting evidence, counterexamples, and actual proofs. IAS G.8.7: Construct logical arguments, judge their validity, and give counterexamples to disprove statements. There is partial alignment with IAS G.8.6 and G.8.7. There is an emphasis on relationships within axiomatic systems and on understanding the differences between supporting evidence, counterexamples, and proofs. Logic and Proofs G.LP.2: Know precise definitions for angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, and plane. Use standard geometric notation. G.LP.3: State, use, and examine the validity of the converse, inverse, and contrapositive of conditional ( if then ) and bi-conditional ( if and only if ) statements. IAS G.8.5: State, use, and examine the validity of the converse, inverse, and contrapositive of if then statements IAS G.8.4: Write and interpret statements of the form if then and if and only if. Bi-conditional ( if and only if ) statements have been added to what students previously were required to understand with IAS G.8.5 and G.8.4. G.LP.4: Develop geometric proofs, including direct proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry, using twocolumn, paragraphs, and flow charts formats. IAS G.8.8: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs. Direct and indirect proofs have been added to the various types of proofs that were already required with IAS G.8.8. Indianapolis Public Schools Curriculum and Instruction Page 1 of 12
G.PL.1: Identify, justify, and apply properties of planes. G.PL.2: Describe the intersection of two or more geometric figures in the same plane. Points, Lines, Angles, and Planes G.PL.3: Prove and apply theorems about lines and angles, including the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent; when a transversal crosses parallel lines, same side interior angles are supplementary; and points on a perpendicular bisector of a line segment are exactly those equidistant from the endpoints of the segment. G.PL.4: Know that parallel lines have the same slope and perpendicular lines have opposite reciprocal slopes. Determine if a pair of lines are parallel, perpendicular, or neither by comparing the slopes in coordinate graphs and in equations. Find the equation of a line, passing through a given point, that is parallel or perpendicular to a given line. G.PL.5: Explain and justify the process used to construct, with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.), congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, and parallel and perpendicular lines. IAS G.1.3: Understand and use the relationships between special pairs of angles formed by parallel lines and transversals. IAS G.1.4: Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines. IAS G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used. IAS G.4.2: Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors. The vague reference to the relationships between special pairs of angles formed in IAS G.1.3 has been replaced by more specific examples outlined clearly in G.PL. 3. There is partial alignment with IAS G.1.4. Students are required to know characteristics of slopes of parallel and perpendicular lines. Additionally, students are required to find the equations of lines parallel and perpendicular to lines with given information. The information contained in IAS G.1.2, G.4.2, and G.8.9 has been streamlined into G.PL.5. IAS G.8.9: Perform basic constructions, describing and justifying the procedures used. Distinguish between constructing and drawing geometric figures. Indianapolis Public Schools Curriculum and Instruction Page 2 of 12
G.T.1: Prove and apply theorems about triangles, including the following: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem, using triangle similarity; and the isosceles triangle theorem and its converse. IAS G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and area. IAS G.4.5: Prove and apply theorems involving segments divided proportionally. IAS G.5.1: Prove and use the Pythagorean Theorem. Students are expected to prove the following: Measure of interior angles of a triangle sum Base angles of isosceles triangles are congruent Segment joining midpoints of two sides of triangle is parallel to third side and half the length Medians of a triangle meet at a point Line parallel to one side of triangle divides other two proportionally Pythagorean Theorem using similarity Isosceles triangle theorem and its converse G.T.2: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Triangles G.T.3: Explain and justify the process used to construct congruent triangles with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). IAS G.4.3: Construct triangles congruent to given triangles. There is partial alignment with IAS G.4.3. Additionally, students must be able to explain and justify the process used when constructing. Specific tools and methods are listed as well. G.T.4: Given two triangles, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides, and to establish the AA criterion for two triangles to be similar. G.T.5: Use properties of congruent and similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles. IAS G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. Indianapolis Public Schools Curriculum and Instruction Page 3 of 12
G.T.6: Prove and apply the inequality theorems, including the following: triangle inequality, inequality in one triangle, and the hinge theorem and its converse. G.T.7: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Understand and use the geometric mean to solve for missing parts of triangles. IAS G.4.8: Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and the hinge theorem. IAS G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. The language of IAS G.4.8 and G.T.6 is very similar. There is partial alignment with IAS G.5.2. Additionally, students are required to understand and use the geometric mean to solve for missing parts of triangles. Triangles G.T.8: Develop the distance formula using the Pythagorem Theorem. Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems. Find measures of the sides of polygons in the coordinate plane; apply this technique to compute the perimeters and areas of polygons in real-world and mathematical problems. G.T.9: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. IAS G.1.1: Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems. IAS G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. There is partial alignment with IAS G.1.1. Additionally, students are required to develop the distance formula using the Pythagorean Theorem. They must also find the measures of sides of polygons and compute perimeters and areas of polygons in real-world situations. G.T.9 references more skills associated with trigonometric ratios. G.T.10: Use trigonometric ratios (sine, cosine and tangent) and the Pythagorean Theorem to solve realworld and mathematical problems involving right triangles. IAS G.5.4: Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. IAS G.5.6: Solve word problems involving right triangles. The skills required in IAS G.5.4 and G.5.6 have merged into standard G.T.10. G.T.11: Use special right triangles (30-60 and 45-45 ) to solve real-world and mathematical problems. IAS G.5.3: Use special right triangles (30-60 and 45-45 ) to solve problems. The phrase real-world and mathematical problems denotes an emphasis on application. Indianapolis Public Schools Curriculum and Instruction Page 4 of 12
G.QP.1: Prove and apply theorems about parallelograms, including the following: opposite sides are congruent; opposite angles are congruent; the diagonals of a parallelogram bisect each other; and rectangles are parallelograms with congruent diagonals. IAS G.3.2: Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas. In addition to using properties to solve problems (as referenced in IAS G.3.2), students must also prove theorems about parallelograms. G.QP.2: Prove that given quadrilaterals are parallelograms, rhombuses, rectangles, squares or trapezoids. Include coordinate proofs of quadrilaterals in the coordinate plane. IAS G.3.1: Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite. There is partial alignment with IAS G.3.1, G.2.6, and G.3.4. The emphasis of G.QP.2 is on proving that given quadrilaterals fit the requirements of various shapes. Quadrilaterals and Polygons G.QP.3: Find measures of interior and exterior angles of polygons. Explain and justify the method used. IAS G.2.6: Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity. IAS G.3.4: Use coordinate geometry to prove properties of quadrilaterals, such as regularity, congruence, and similarity. IAS G.2.2: Find measures of interior and exterior angles of polygons, justifying the method used. The language of IAS G.2.2 and G.QP.3 is similar. G.QP.4: Identify types of symmetry of polygons, including line, point, rotational, and self-congruencies. G.QP.5: Deduce formulas relating lengths and sides, perimeters, and areas of regular polygons. Understand how limiting cases of such formulas lead to expressions for the circumference and the area of a circle. IAS G.2.5: Find and use measures of sides, perimeters, and areas of polygons. Relate these measures to each other using formulas. There is an emphasis on students deducing formulas pertaining to polygons. Additionally, they must understand how limiting cases of these formulas leads to expressions pertaining to circles. IAS G.3.3: Find and use measures of sides, perimeters, and areas of quadrilaterals. Relate these measures to each other using formulas. Indianapolis Public Schools Curriculum and Instruction Page 5 of 12
G.CI.1: Define, identify and use relationships among the following: radius, diameter, arc, measure of an arc, chord, secant, tangent, and congruent concentric circles. IAS G.6.2: Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent. IAS G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). The skills mentioned in IAS G.6.2, G.6.5, and G.6.6 have merged into the language of G.CI.1. The skills mentioned in G.6.1 and G.6.7 will help students and their conceptual knowledge pertaining to G.CI.1. IAS G.6.6: Define and identify congruent and concentric circles. Circles G.CI.2: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. IAS G.6.7: Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems. An emphasis is placed on deriving the formula for the area of a sector and on the fact that the length of the arc intercepted by an angle is proportional to the radius. G.CI.3: Identify and describe relationships among inscribed angles, radii, and chords, including the following: the relationship that exists between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; and the radius of a circle is perpendicular to a tangent where the radius intersects the circle. IAS G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). The emphasis of G.CI.3 is on identifying and describing the listed relationships. G.CI.4: Solve real-world and other mathematical problems that involve finding measures of circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed, and intersections of secants and tangents). IAS G.6.5: Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). The emphasis of G.CI.4 is on solving real-world and other mathematical problems involving the listed items. Indianapolis Public Schools Curriculum and Instruction Page 6 of 12
G.CI.5: Construct a circle that passes through three given points not on a line and justify the process used. IAS G.6.1: Find the center of a given circle. Construct the circle that passes through three given points not on a line. Circles G.CI.6: Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. IAS G.6.4: Construct tangents to circles and circumscribe and inscribe circles. There is partial alignment with IAS G.6.4. Additionally, there is an emphasis placed on students justifying the process used for these constructions. G.CI.7: Construct the inscribed and circumscribed circles of a triangle with or without technology, and prove properties of angles for a quadrilateral inscribed in a circle. IAS G.6.4: Construct tangents to circles and circumscribe and inscribe circles. Students must be able to construct both with and without technology and must prove properties of angles for a quadrilateral inscribed in a circle. G.TR.1: Use geometric descriptions of rigid motions to transform figures and to predict and describe the results of translations, reflections and rotations on a given figure. Describe a motion or series of motions that will show two shapes are congruent. IAS G.2.4: Apply transformations (slides, flips, turns, expansions, and contractions) to polygons to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips, and turns are congruent to the original shape. There is partial alignment with IAS G.2.4. Students must be able to describe the series of motions that will result in two congruent shapes. Transformations G.TR.2: Understand a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. Indianapolis Public Schools Curriculum and Instruction Page 7 of 12
G.TS.1: Describe relationships between the faces, edges, and vertices of three-dimensional solids. Create a net for a given three-dimensional solid. Describe the three-dimensional solid that can be made from a given net (or pattern). IAS G.7.3: Describe relationships between the faces, edges, and vertices of polyhedra. There is partial alignment with IAS G.7.3. Additionally, students will create nets for the given 3-D solids and describe how a solid can be made from the given net. G.TS.2: Describe symmetries of three-dimensional solids. IAS G.7.4: Describe symmetries of geometric solids. Three-dimensional solids are specifically referenced in G.TS.2. Three-Dimensional Solids G.TS.3: Know properties of congruent and similar solids, including prisms, regular pyramids, cylinders, cones, and spheres; solve problems involving congruent and similar solids. G.TS.4: Describe sets of points on spheres, including chords, tangents, and great circles. G.TS.5: Solve real-world and other mathematical problems involving volume and surface area of prisms, cylinders, cones, spheres, and pyramids, including problems that involve algebraic expressions. IAS G.7.6: Identify and know properties of congruent and similar solids. IAS G.7.5: Describe sets of points on spheres: chords, tangents, and great circles. IAS G.7.7: Find and use measures of sides, volumes of solids, and surface areas of solids. Relate these measures to each other using formulas. Specific solids are referenced in the language of G.TS.3. There is partial alignment with IAS G.7.7 There is an emphasis on real-world problems and problems involving algebraic expressions. G.TS.6: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). G.TS.7: Graph points on a three-dimensional coordinate plane. Explain how the coordinates relate the point as the distance from the origin on each of the three axes. Indianapolis Public Schools Curriculum and Instruction Page 8 of 12
G.TS.8: Determine the distance of a point to the origin on the three-dimensional coordinate plane using the distance formula. G.TS.9: Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects Indianapolis Public Schools Curriculum and Instruction Page 9 of 12
PROCESS STANDARDS FOR MATHEMATICS The Process Standards demonstrate the ways in which students should develop conceptual understanding of mathematical content, and the ways in which students should synthesize and apply mathematical skills. PS.1: Make sense of problems and persevere in solving them. PS.2: Reason abstractly and quantitatively. PS.3: Construct viable arguments and critique the reasoning of others. PROCESS STANDARDS FOR MATHEMATICS Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway, rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? and "Is my answer reasonable?" They understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand how mathematical ideas interconnect and build on one another to produce a coherent whole. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They analyze situations by breaking them into cases and recognize and use counterexamples. They organize their mathematical thinking, justify their conclusions and communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. They justify whether a given statement is true always, sometimes, or never. Mathematically proficient students participate and collaborate in a mathematics community. They listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Indianapolis Public Schools Curriculum and Instruction Page 10 of 12
PS.4: Model with mathematics. PS.5: Use appropriate tools strategically. PS.6: Attend to precision. PS.7: Look for and make use of structure. PS.8: Look for and express regularity in repeated reasoning. Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace using a variety of appropriate strategies. They create and use a variety of representations to solve problems and to organize and communicate mathematical ideas. Mathematically proficient students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Mathematically proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students identify relevant external mathematical resources, such as digital content, and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and to support the development of learning mathematics. They use technology to contribute to concept development, simulation, representation, reasoning, communication and problem solving. Mathematically proficient students communicate precisely to others. They use clear definitions, including correct mathematical language, in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They express solutions clearly and logically by using the appropriate mathematical terms and notation. They specify units of measure and label axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and check the validity of their results in the context of the problem. They express numerical answers with a degree of precision appropriate for the problem context. Mathematically proficient students look closely to discern a pattern or structure. They step back for an overview and shift perspective. They recognize and use properties of operations and equality. They organize and classify geometric shapes based on their attributes. They see expressions, equations, and geometric figures as single objects or as being composed of several objects. Mathematically proficient students notice if calculations are repeated and look for general methods and shortcuts. They notice regularity in mathematical problems and their work to create a rule or formula. Mathematically proficient students maintain oversight of the process, while attending to the details as they solve a problem. They continually evaluate the reasonableness of their intermediate results. Indianapolis Public Schools Curriculum and Instruction Page 11 of 12
Indicators that No Longer Need to Be Taught In This Course Content Standard Comments IAS G.2.1: Identify and describe convex, concave, and regular polygons. No longer taught in Indiana Academic Standards 2014 IAS G.4.1: Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. IAS G.4.7: Find and use measures of sides, perimeters, and areas of triangles. Relate these measures to each other using formulas. No longer taught in Indiana Academic Standards 2014 No longer taught in Indiana Academic Standards 2014 IAS G.5.5: Know and use the relationship sin 2 x + cos 2 x = 1. No longer taught in Indiana Academic Standards 2014 IAS G.6.8: Find the equation of a circle in the coordinate plane in terms of its center and radius. No longer taught in Indiana Academic Standards 2014 IAS G.7.1: Describe and make regular and nonregular polyhedra. No longer taught in Indiana Academic Standards 2014 IAS G.7.2: Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron. IAS G.8.1: Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards. No longer taught in Indiana Academic Standards 2014 No longer taught in Indiana Academic Standards 2014 IAS G.8.2: Decide whether a solution is reasonable in the context of the original situation. No longer taught in Indiana Academic Standards 2014 IAS G.8.3: Make conjectures about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. No longer taught in Indiana Academic Standards 201 Indianapolis Public Schools Curriculum and Instruction Page 12 of 12